Title: Epimorphisms and boundary slopes of 2–bridge knots
Abstract: In this article we study a partial ordering on knots in S 3 where K 1 K 2 if there is an epimorphism from the knot group of K 1 onto the knot group of K 2 which preserves peripheral structure.If K 1 is a 2-bridge knot and K 1 K 2 , then it is known that K 2 must also be 2-bridge.Furthermore, Ohtsuki, Riley and Sakuma give a construction which, for a given 2-bridge knot K p=q , produces infinitely many 2-bridge knots K p 0 =q 0 with K p 0 =q 0 K p=q .After characterizing all 2-bridge knots with 4 or less distinct boundary slopes, we use this to prove that in any such pair, K p 0 =q 0 is either a torus knot or has 5 or more distinct boundary slopes.We also prove that 2-bridge knots with exactly 3 distinct boundary slopes are minimal with respect to the partial ordering.This result provides some evidence for the conjecture that all pairs of 2-bridge knots with K p 0 =q 0 K p=q arise from the Ohtsuki-Riley-Sakuma construction.